Exact reverse-mathematical strength of CTL VEL

Determine the exact reverse-mathematical strength of the Valuation Extension Lemma for Computation Tree Logic (CTL), currently known to lie between \(\Sigma^1_2\text{-}\mathrm{DC}_0\) and \(\Pi^1_1\text{-}\mathrm{CA}_0\) over \(\mathrm{RCA}_0\).

Background

The paper establishes that VEL for CTL follows from Σ21-DC0\Sigma^1_2\text{-}\mathrm{DC}_0 and implies Π11-CA0\Pi^1_1\text{-}\mathrm{CA}_0, leaving a gap between these systems. The authors note that VEL for CTL is expressible as a Π31\Pi^1_3 sentence and explain that an implication from Σ21-AC0\Sigma^1_2\text{-}\mathrm{AC}_0 would, using a conservativity result, yield equivalence with Π11-CA0\Pi^1_1\text{-}\mathrm{CA}_0 over RCA0\mathrm{RCA}_0.

References

On the other hand, the reverse-mathematical analysis of VEL for $\mathbf{CTL}$ and $\mathbf{LTL}$ is not yet complete. VEL for $\mathbf{CTL}$ lies between $\Sigma1_2$-$\mathrm{DC}_0$ and $\Pi1_1$-$\mathrm{CA}_0$, but its exact position is still unknown.

— Frame definability in second-order arithmetic  (2608.22822 - Takeda, 24 Aug 2026) in Section “Conclusion and Future Research”

Does $\Sigma1_2\text{-$\mathrm{AC}_0$}$ imply (Restricted) VEL for $\mathbf{CTL}$?

— Frame definability in second-order arithmetic  (2608.22822 - Takeda, 24 Aug 2026) in Question environment in Section “Conclusion and Future Research”